\(\int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx\) [249]

Optimal result

Integrand size = 18, antiderivative size = 211 \[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=-\frac {(1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}}{(1+i a) x}-\frac {3 i b \arctan \left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{7/4} \sqrt [4]{i+a}}-\frac {3 i b \text {arctanh}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i a+i b x}}{\sqrt [4]{i-a} \sqrt [4]{1-i a-i b x}}\right )}{(i-a)^{7/4} \sqrt [4]{i+a}} \] Output:

-(1-I*a-I*b*x)^(3/4)*(1+I*a+I*b*x)^(1/4)/(1+I*a)/x-3*I*b*arctan((I+a)^(1/4 
)*(1+I*a+I*b*x)^(1/4)/(I-a)^(1/4)/(1-I*a-I*b*x)^(1/4))/(I-a)^(7/4)/(I+a)^( 
1/4)-3*I*b*arctanh((I+a)^(1/4)*(1+I*a+I*b*x)^(1/4)/(I-a)^(1/4)/(1-I*a-I*b* 
x)^(1/4))/(I-a)^(7/4)/(I+a)^(1/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.51 \[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=-\frac {(-i (i+a+b x))^{3/4} \left (1+a^2+i b x+a b x-2 i b x \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},\frac {1+a^2-i b x+a b x}{1+a^2+i b x+a b x}\right )\right )}{\left (1+a^2\right ) x (1+i a+i b x)^{3/4}} \] Input:

Integrate[1/(E^(((3*I)/2)*ArcTan[a + b*x])*x^2),x]
 

Output:

-((((-I)*(I + a + b*x))^(3/4)*(1 + a^2 + I*b*x + a*b*x - (2*I)*b*x*Hyperge 
ometric2F1[3/4, 1, 7/4, (1 + a^2 - I*b*x + a*b*x)/(1 + a^2 + I*b*x + a*b*x 
)]))/((1 + a^2)*x*(1 + I*a + I*b*x)^(3/4)))
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5618, 105, 104, 756, 218, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx\)

\(\Big \downarrow \) 5618

\(\displaystyle \int \frac {(-i a-i b x+1)^{3/4}}{x^2 (i a+i b x+1)^{3/4}}dx\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {3 b \int \frac {1}{x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}dx}{2 (-a+i)}-\frac {(-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}{(1+i a) x}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {6 b \int \frac {1}{-i a+\frac {(1-i a) (i a+i b x+1)}{-i a-i b x+1}-1}d\frac {\sqrt [4]{i a+i b x+1}}{\sqrt [4]{-i a-i b x+1}}}{-a+i}-\frac {(-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}{(1+i a) x}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {6 b \left (-\frac {i \int \frac {1}{\sqrt {i-a}-\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-i a-i b x+1}}}d\frac {\sqrt [4]{i a+i b x+1}}{\sqrt [4]{-i a-i b x+1}}}{2 \sqrt {-a+i}}-\frac {i \int \frac {1}{\sqrt {i-a}+\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-i a-i b x+1}}}d\frac {\sqrt [4]{i a+i b x+1}}{\sqrt [4]{-i a-i b x+1}}}{2 \sqrt {-a+i}}\right )}{-a+i}-\frac {(-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}{(1+i a) x}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {6 b \left (-\frac {i \int \frac {1}{\sqrt {i-a}-\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-i a-i b x+1}}}d\frac {\sqrt [4]{i a+i b x+1}}{\sqrt [4]{-i a-i b x+1}}}{2 \sqrt {-a+i}}-\frac {i \arctan \left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{2 (-a+i)^{3/4} \sqrt [4]{a+i}}\right )}{-a+i}-\frac {(-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}{(1+i a) x}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {6 b \left (-\frac {i \arctan \left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{2 (-a+i)^{3/4} \sqrt [4]{a+i}}-\frac {i \text {arctanh}\left (\frac {\sqrt [4]{a+i} \sqrt [4]{i a+i b x+1}}{\sqrt [4]{-a+i} \sqrt [4]{-i a-i b x+1}}\right )}{2 (-a+i)^{3/4} \sqrt [4]{a+i}}\right )}{-a+i}-\frac {(-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}{(1+i a) x}\)

Input:

Int[1/(E^(((3*I)/2)*ArcTan[a + b*x])*x^2),x]
 

Output:

-(((1 - I*a - I*b*x)^(3/4)*(1 + I*a + I*b*x)^(1/4))/((1 + I*a)*x)) + (6*b* 
(((-1/2*I)*ArcTan[((I + a)^(1/4)*(1 + I*a + I*b*x)^(1/4))/((I - a)^(1/4)*( 
1 - I*a - I*b*x)^(1/4))])/((I - a)^(3/4)*(I + a)^(1/4)) - ((I/2)*ArcTanh[( 
(I + a)^(1/4)*(1 + I*a + I*b*x)^(1/4))/((I - a)^(1/4)*(1 - I*a - I*b*x)^(1 
/4))])/((I - a)^(3/4)*(I + a)^(1/4))))/(I - a)
 

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 5618

Int[E^(ArcTan[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), 
 x_Symbol] :> Int[(d + e*x)^m*((1 - I*a*c - I*b*c*x)^(I*(n/2))/(1 + I*a*c + 
 I*b*c*x)^(I*(n/2))), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
 

Maple [F]

\[\int \frac {1}{{\left (\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )}^{\frac {3}{2}} x^{2}}d x\]

Input:

int(1/((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)/x^2,x)
 

Output:

int(1/((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)/x^2,x)
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (137) = 274\).

Time = 0.14 (sec) , antiderivative size = 613, normalized size of antiderivative = 2.91 \[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\frac {3 \, \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (-i \, a - 1\right )} x \log \left (\frac {b \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} + \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (a^{2} - 2 i \, a - 1\right )}}{b}\right ) + 3 \, \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (i \, a + 1\right )} x \log \left (\frac {b \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (a^{2} - 2 i \, a - 1\right )}}{b}\right ) - 3 \, \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (a - i\right )} x \log \left (\frac {b \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (i \, a^{2} + 2 \, a - i\right )}}{b}\right ) + 3 \, \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (a - i\right )} x \log \left (\frac {b \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - \left (-\frac {b^{4}}{a^{8} - 6 i \, a^{7} - 14 \, a^{6} + 14 i \, a^{5} + 14 i \, a^{3} + 14 \, a^{2} - 6 i \, a - 1}\right )^{\frac {1}{4}} {\left (-i \, a^{2} - 2 \, a + i\right )}}{b}\right ) + 2 \, {\left (b x + a + i\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{2 \, {\left (a - i\right )} x} \] Input:

integrate(1/((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)/x^2,x, algorithm="fr 
icas")
 

Output:

1/2*(3*(-b^4/(a^8 - 6*I*a^7 - 14*a^6 + 14*I*a^5 + 14*I*a^3 + 14*a^2 - 6*I* 
a - 1))^(1/4)*(-I*a - 1)*x*log((b*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) 
/(b*x + a + I)) + (-b^4/(a^8 - 6*I*a^7 - 14*a^6 + 14*I*a^5 + 14*I*a^3 + 14 
*a^2 - 6*I*a - 1))^(1/4)*(a^2 - 2*I*a - 1))/b) + 3*(-b^4/(a^8 - 6*I*a^7 - 
14*a^6 + 14*I*a^5 + 14*I*a^3 + 14*a^2 - 6*I*a - 1))^(1/4)*(I*a + 1)*x*log( 
(b*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)) - (-b^4/(a^8 - 
6*I*a^7 - 14*a^6 + 14*I*a^5 + 14*I*a^3 + 14*a^2 - 6*I*a - 1))^(1/4)*(a^2 - 
 2*I*a - 1))/b) - 3*(-b^4/(a^8 - 6*I*a^7 - 14*a^6 + 14*I*a^5 + 14*I*a^3 + 
14*a^2 - 6*I*a - 1))^(1/4)*(a - I)*x*log((b*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x 
+ a^2 + 1)/(b*x + a + I)) - (-b^4/(a^8 - 6*I*a^7 - 14*a^6 + 14*I*a^5 + 14* 
I*a^3 + 14*a^2 - 6*I*a - 1))^(1/4)*(I*a^2 + 2*a - I))/b) + 3*(-b^4/(a^8 - 
6*I*a^7 - 14*a^6 + 14*I*a^5 + 14*I*a^3 + 14*a^2 - 6*I*a - 1))^(1/4)*(a - I 
)*x*log((b*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)) - (-b^4 
/(a^8 - 6*I*a^7 - 14*a^6 + 14*I*a^5 + 14*I*a^3 + 14*a^2 - 6*I*a - 1))^(1/4 
)*(-I*a^2 - 2*a + I))/b) + 2*(b*x + a + I)*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + 
 a^2 + 1)/(b*x + a + I)))/((a - I)*x)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\text {Timed out} \] Input:

integrate(1/((1+I*(b*x+a))/(1+(b*x+a)**2)**(1/2))**(3/2)/x**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\int { \frac {1}{x^{2} \left (\frac {i \, b x + i \, a + 1}{\sqrt {{\left (b x + a\right )}^{2} + 1}}\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)/x^2,x, algorithm="ma 
xima")
 

Output:

integrate(1/(x^2*((I*b*x + I*a + 1)/sqrt((b*x + a)^2 + 1))^(3/2)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)/x^2,x, algorithm="gi 
ac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Warning, need to choose a branch fo 
r the root of a polynomial with parameters. This might be wrong.The choice 
 was done
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\int \frac {1}{x^2\,{\left (\frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{\sqrt {{\left (a+b\,x\right )}^2+1}}\right )}^{3/2}} \,d x \] Input:

int(1/(x^2*((a*1i + b*x*1i + 1)/((a + b*x)^2 + 1)^(1/2))^(3/2)),x)
 

Output:

int(1/(x^2*((a*1i + b*x*1i + 1)/((a + b*x)^2 + 1)^(1/2))^(3/2)), x)
 

Reduce [F]

\[ \int \frac {e^{-\frac {3}{2} i \arctan (a+b x)}}{x^2} \, dx=\int \frac {1}{{\left (\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )}^{\frac {3}{2}} x^{2}}d x \] Input:

int(1/((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)/x^2,x)
 

Output:

int(1/((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)/x^2,x)